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Difficulty: MediumEnergy Levels and Atomic Spectra

An electron in a hydrogen atom drops from an excited state with energy −0.85 eV-0.85\text{ eV} to a lower energy state of −3.40 eV-3.40\text{ eV}. What is the wavelength of the photon emitted during this transition? (h=6.6×10−34 J⋅sh = 6.6 \times 10^{-34}\text{ J}\cdot\text{s}, c=3.0×108 m/sc = 3.0 \times 10^8\text{ m/s}, 1 eV=1.6×10−19 J1\text{ eV} = 1.6 \times 10^{-19}\text{ J})

  1. 4.85×10−7 m4.85 \times 10^{-7}\text{ m}Answer
  2. B
    2.91×10−7 m2.91 \times 10^{-7}\text{ m}
  3. C
    7.76×10−26 m7.76 \times 10^{-26}\text{ m}
  4. D
    6.18×1014 m6.18 \times 10^{14}\text{ m}

Answer

The wavelength of the emitted photon is 4.85×10−7 m4.85 \times 10^{-7}\text{ m}.
The energy of the emitted photon is given by ΔE=−0.85 eV−(−3.40 eV)=2.55 eV\Delta E = -0.85\text{ eV} - (-3.40\text{ eV}) = 2.55\text{ eV}. Converting to Joules gives 2.55×1.6×10−19 J=4.08×10−19 J2.55 \times 1.6 \times 10^{-19}\text{ J} = 4.08 \times 10^{-19}\text{ J}. Substituting into λ=hcΔE\lambda = \frac{hc}{\Delta E} yields λ=6.6×10−34×3.0×1084.08×10−19=4.85×10−7 m\lambda = \frac{6.6 \times 10^{-34} \times 3.0 \times 10^8}{4.08 \times 10^{-19}} = 4.85 \times 10^{-7}\text{ m}.

Step-by-Step Solution

1
Calculate the energy difference between the initial and final states
ΔE=Ei−Ef=−0.85 eV−(−3.40 eV)=2.55 eV\Delta E = E_i - E_f = -0.85\text{ eV} - (-3.40\text{ eV}) = 2.55\text{ eV}
The energy of the emitted photon equals the energy difference between the two levels.
2
Convert the energy difference from electron-volts to Joules
ΔE=2.55×1.6×10−19 J=4.08×10−19 J\Delta E = 2.55 \times 1.6 \times 10^{-19}\text{ J} = 4.08 \times 10^{-19}\text{ J}
Standard SI units are required for calculations involving Planck's constant.
3
Calculate photon wavelength using λ=hcΔE\lambda = \frac{hc}{\Delta E}
λ=(6.6×10−34 J⋅s)(3.0×108 m/s)4.08×10−19 J=4.85×10−7 m\lambda = \frac{(6.6 \times 10^{-34}\text{ J}\cdot\text{s})(3.0 \times 10^8\text{ m/s})}{4.08 \times 10^{-19}\text{ J}} = 4.85 \times 10^{-7}\text{ m}
Relating photon energy to wavelength using Planck's constant and speed of light.

Key Concept

Photon Emission and Atomic Transitions
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Topics

Energy Levels and Atomic Spectra

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